The key pieces for the implementation of the multifidelity Monte Carlo simulation are the calculation of the parameters α∗ and r∗ and an iterative procedure to incre-ment the number of samples. We also discuss practical aspects including estimating statistics other than the mean and estimating functions of statistics.
2.2.1 Parameter Calculations
In practice, since σA, σB, and ρAB are unknown, the optimal parameters α∗ and r∗ are replaced by their sample estimates ˆα and ˆr based on the n samples of A(ω) and B(ω), {ai, bi}ni=1:
To assess the impact of the errors in ˆα and ˆr relative to the exact parameters α∗ and r∗, we plot the ratio less than one indicates a reduction in variance over the regular Monte Carlo estimator at the same computational effort. It can be seen that there is reasonable room for deviations from α∗ and r∗ (location indicated by the cross), although optimal variance reduction will not be achieved.
Control Parameter
Ratio of Number of Model Evaluations
1
(a) Correlation Coefficient ρAB = 0.9
Control Parameter
Ratio of Number of Model Evaluations
1
(b) Correlation Coefficient ρAB = 0.95 Figure 2-2: Contour plots of Var [ˆsA,p]/ Var [¯ap] as a function of control parameter α and ratio of the number of model evaluations r with w = 30 and σA/σB = 1. The cross indicates the location of (α∗, r∗).
2.2.2 Iterative Procedure
The procedure to compute the multifidelity estimator begins with a set of n = ninit
samples {ai, bi}ni=1 that is incremented by n∆ every iteration. The computation times for the initial ninit samples can be used to determine w, if not already available, and n∆may be chosen based on the number of processors available for parallel evaluations.
During each iteration, the samples {ai, bi}ni=1 are used to calculate the parameters ˆα and ˆr. This then determines the total number of low-fidelity model evaluations needed to compute the multifidelity estimator (including the existing n low-fidelity model evaluations) as m = nˆr. Omitting algorithm overhead, the computational expense is therefore p = n + m/w. An algorithm to compute the multifidelity estimator ˆsA,p for sA = E [A(ω)] = E [Mhigh(U(ω))] is shown in Algorithm 2.1. We emphasize that all of the samples used in the algorithm are generated from the same stream of random input vectors ui, i = 1, 2, 3, . . . in order to induce the correlation needed for the multifidelity estimator.
Algorithm 2.1 Multifidelity Estimator
Given desired RMSE, ratio of average computation time w, initial number of samples ninit, increment in number of samples n∆, high-fidelity model Mhigh(u), low-fidelity model Mlow(u), and a sequence of pseudo-random input vectors ui for i = 1, 2, 3, . . . drawn from the distribution of U(ω):
1 Let nold = 0, mold= 0, ` = 0, and n = ninit.
9 Compute multifidelity estimator from (2.2).
10 Compute RMSE from (2.3).
11 If RMSE is too large, set nold= n, mold= m, n ← n + n∆and return to Step 2;
otherwise, stop.
2.2.3 Estimating Variance
The preceding development assumes that we are interested in estimating sA= E [A(ω)] = E [Mhigh(U(ω))]. We may also use the procedure to estimate other statistics such as sA = Var [A(ω)] = Var [Mhigh(U(ω))], but some care is needed to ensure that the method is efficient. For example, computing the variance based on the formula EA(ω)2 − (E [A(ω)])2 can produce unsatisfactory results because ρA2B2 tends to be worse than ρAB. Instead, it is usually better to define the samples from the residuals as
based on the one-pass algorithm for computing variance [28]. Using the above
and so we can apply the approach described in §2.1.2 to the redefined samples {ai, bi} as if we were estimating the mean. The correlation coefficient based on the residuals is usually higher than that based on the square of the random variables.2 However, ai and bi are no longer i.i.d. samples and the theory of §2.1.2 is not valid. Neverthe-less, the results in §2.4 suggest that this approach is still effective in estimating the variance.
We can similarly redefine the samples to estimate other quantities of interest, e.g., indicator functions of the model outputs to estimate probabilities. More advanced control variate techniques are available to estimate quantiles [20]. However, special-ized methods may be needed to efficiently estimate rare probabilities and reliability metrics [30], since the control variate approach does not directly address the problem of landing few samples in the rare event region.
2.2.4 Estimating Functions of Statistics
In optimization under uncertainty, we may be concerned with functions of one or more statistics. For example, a robust objective function may be formulated as E [Mhigh(U(ω))] +pVar [Mhigh(U(ω))]. Let sA be the q × 1 vector of statistics of interest (e.g., q = 2 and sA = [E [Mhigh(U(ω))] Var [Mhigh(U(ω))]]>) and let ˆsA,p be the q × 1 vector of its estimator. The function may then be written as f (sA) and we estimate it as f (ˆsA,p).
The error in the function estimator f (ˆsA,p) can be approximated using a first-order
2For example, consider a low-fidelity model that is simply a constant offset of the high-fidelity model.
Taylor expansion about sA:
f (ˆsA,p) − f (sA) ≈ ∇sf (ˆsA,p)>(ˆsA,p− sA).
Squaring both sides and taking the expectation, we obtain an approximation of the mean square error [52]
MSE [f (ˆsA,p)] ≈ ∇sf (ˆsA,p)>Cov [ˆsA,p]∇sf (ˆsA,p), (2.5)
where Cov [ˆsA,p] is the q × q covariance matrix of the vector of estimators ˆsA,p. Thus, we need to generalize the scalar multifidelity estimator ˆsA,p in (2.2) to the vector case ˆsA,p.
Let A(ω) and B(ω) be q × 1 random vectors and let ΣA = Cov [A(ω)], ΣB = Cov [B(ω)], and ΣAB = Cov [A(ω), B(ω)] be their q × q covariance matrices and cross-covariance matrix. Also, let α be a q × q diagonal matrix whose elements are the q (optimal) control parameters for each of the q components of ˆsA,p. Then, the q × 1 vector of multifidelity estimators is
ˆsA,p = ¯an+ α ¯bm− ¯bn
and the q × q covariance matrix of the vector of multifidelity estimators is
Cov [ˆsA,p] = Cov [¯an] + α Covb¯mα>+ α Covb¯nα>
The covariance matrix of the vector of multifidelity estimators can then be used to compute the approximation of the mean square error of the function estimator in (2.5). This error estimate is useful for predicting the number of samples needed to control the objective function noise during optimization.